## Barrycentric coordinates for a polytope

For the past few weeks, I have been searching about this topic: Suppose we are given a convex polytope having vertices say $$A_1, A_2,...,A_n$$ where each $$A_i, i=1,...,n$$ represent a matrix - Infact its a convex polytope whose vertices are matrices. How can we verify if a given a matrix say $$A_t$$ can be written as a convex combination of the vertices.

To elaborate more: I am constructing this polytope to encompass a time-varying matrix; say denoted by A(t) where the variable "t" is varying between some upper and lower bounds t_{min} and t_{max}. Using these bounds of t, I find vertices for a polytope denoted by $$A_1,...,A_n$$. Now how do I prove that given A(t) where $$t_min \leq t \leq t_max$$ can be written as convex combination of the vertices $$A_1,...,A_n$$. Any help would be appreciated.

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 Tags convex, matrices, polytopes